My two cents:
As a common practice, all the clans may be required to send someone from each family to court. It can be done with younger/eldest sons/daughters, however you like it. This is a nice way of saying, "you rebel against me, your son's head is forfait".
Of course, it may happen that a family still will want to rebel, at the risk of endangering their hostages. It doesn't matter: as long as you have the approval and the support of the other families, including your own, you should be able, as a ruler, to deal with possible treason.
To keep its rule, the new ruler should know quite well the inner workings of his clan ... but he/she should know the other quite as good. It doesn't matter if this is achieved through study or espionage. The point is: there are key figures in each clan that can be bought over by the right prize.
You can play this in a number of ways. It makes sense - as you described - that a ruler will appoint his champion as head of the military (as who controls the army controls a fair share of power).
But what about other roles? There are other jobs both high on importance and status. The ruler could appoint a clan's head as the master of coin, for example. Other things might be of importance - such as land, rights on commerce, lower taxes, and so on. The point is: the ruler should be careful not to advantage his own clan too much in spite of the others.
Of course politics are a tricky thing, since dealing out too many privileges will result, eventually, with a weaker crown - this has happened a lot through history, I'm thinking mainly about feudal europe after Charle Magne.
Don't piss on everyone's garden
Each major clan in your world is probably somewhat established in its own territory. This means each will have its own castles/cities, source of income, main strenghts and charateristics ... in the previous point, I've tried to explain why the ruler should try make some of the other clans happy.
Now I'm gonna explain why, at least, the ruler should avoid making them angry. Yes, it's as obvious as it sounds.
The point here is that your systems is very convenient for every family. There is no dinasty here, since everyone who wins the turnament can have a go at ruling. Also, the winner is decided with champions. Losing a champions equals to losing one person, that's pretty much better than a full-scale war.
So, the head of a clan must have a pretty good reason to risk conflict. Humans are ambitious, true, but most humans are also reasonable. They won't try to change the state of things if the cons outweight the pros.
And how does the ruler ensures that? Simply enough, avoiding stepping into each clan territory. For example, a clan may take most of its revenue by selling pelts on the markets (maybe they live in the northern part of the kingdom, where wild fauna is more abundant). It would be a bad move to raise taxes on such goods, and could be detrimental on the long run. Another clan could be the owner of an important market/city/strategically placed castle. If not absolutely necessary, the king shouldn't try to take that away.
I'm not saying that a ruler shouldn't change how things run and shouldn't extend his power. I'm saying that, if he/she decides to do so, it must be done with utmost caution. There's nothing more dangerous than the creeping sensation that the king is being unjust towards one specific clan to foster rebellion.
Sure, someone in some clan may want to rule ... but what about his wife? His sons? His relatives? His subordinates? No head of clan can scheme alone. If more or less everyone has no particular reason to rebel, he will probably choose stability over change.
Use the church to your advantage
As opposed to "be used by the church". In your world religion holds power over the monarchy. It wouldn't be religion if it didn't teach some sort of belief ... and you can use that. By coming to pact with the head of the church, the ruler can reinforce the idea that he's the only legitimate king/queen, and that he has every right to rule.
Devouts will believe this, thus going against rebellion. Of course, I'll assume most head of clans will have at least a skeptical view of religion, but this should keep at least the ranks in check.